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Quantized symplectic actions and $W$-algebras


Author: Ivan Losev
Journal: J. Amer. Math. Soc. 23 (2010), 35-59
MSC (2000): Primary 17B35, 53D55
DOI: https://doi.org/10.1090/S0894-0347-09-00648-1
Published electronically: September 18, 2009
MathSciNet review: 2552248
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Abstract: With a nilpotent element in a semisimple Lie algebra $\mathfrak {g}$ one associates a finitely generated associative algebra $\mathcal {W}$ called a $W$-algebra of finite type. This algebra is obtained from the universal enveloping algebra $U(\mathfrak {g})$ by a certain Hamiltonian reduction. We observe that $\mathcal {W}$ is the invariant algebra for an action of a reductive group $G$ with Lie algebra $\mathfrak {g}$ on a quantized symplectic affine variety and use this observation to study $\mathcal {W}$. Our results include an alternative definition of $\mathcal {W}$, a relation between the sets of prime ideals of $\mathcal {W}$ and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of $\mathcal {W}$ in the case of classical $\mathfrak {g}$ and the separation of elements of $\mathcal {W}$ by finite-dimensional representations.


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Additional Information

Ivan Losev
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
MR Author ID: 775766
Email: ivanlosev@math.mit.edu

Keywords: $W$-algebras, nilpotent elements, universal enveloping algebras, deformation quantization, prime ideals, finite-dimensional representations
Received by editor(s): August 17, 2007
Published electronically: September 18, 2009
Article copyright: © Copyright 2009 American Mathematical Society